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\title{实变函数第三章：测度论}
\author{CQX ET AL}

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\begin{frame}{第三章目录 }

\begin{enumerate}

\item[3.1.] 外测度
\item[3.2.] 可测集
\item[3.3.] 可测集类
\item[3.4.] 不可测集

\end{enumerate}

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%\begin{frame}{第三章重点 }
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\begin{frame}{3.0.1.  }

\begin{itemize}

\item  {\color{red}问题：关于长度的概念，包含哪些约定俗成的公理？ }

%\item  解答：


\end{itemize}

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\begin{frame}{3.0.2.  }

\begin{itemize}

\item  {\color{red}问题：勒贝格的测度概念，包含哪些公理？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.1.1. 外测度 }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是 $\mathbb{R}^n$ 的点集。什么是 $E$ 的勒贝格外测度 $m^*(E)$? }

%\item  解答：


\end{itemize}

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\begin{frame}{3.1.2.  }

\begin{itemize}

\item  {\color{red}问题：证明外测度具有下述三条基本性质： }
\begin{enumerate}
\item  $m^*(E)\ge 0$, 且当 $E$ 为空集时，$m^*(E)=0$. 
\item  单调性：设 $A\subseteq B$, 则 $m^*(A)\le m^*(B)$. 
\item  次可数可加性：$m^*(\cup_{n=1}^{\infty} A_i) \le \sum_{n=1}^{\infty}m^*(A_i)$. 
\end{enumerate} 

%\item  解答：


\end{itemize}

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\begin{frame}{3.1.3.  }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是 $[0,1]$ 中的全体有理数。证明 $m^*(E)=0$.  }

%\item  解答：


\end{itemize}

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\begin{frame}{3.1.4.  }

\begin{itemize}

\item  {\color{red}问题：设 $E=(a,b)$ 为直线上的区间。证明 $m^*(E)=b-a$.  }

%\item  解答：


\end{itemize}

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\begin{frame}{3.2.1. 可测集 }

\begin{itemize}

\item  {\color{red}问题：构造直线上的互不相交的一列集合 $E_n$, 使得 
$$m^*\left( \bigcup\limits_{n=1}^{\infty} E_n\right) < \sum_{n=1}^{\infty} m^*(E_n). $$ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.2.2. 引理 }

\begin{itemize}

\item  {\color{red}问题：%记 $\mathcal{M}$ 是实数直线上满足勒贝格测度公理的子集组成的集合族。
设 $E\subseteq \mathbb{R}$ 是任意一个子集。证明下述两个条件等价：}
\begin{enumerate}
\item  {\color{red}对任意开区间 $I\subseteq \mathbb{R}$ 都成立 $m^*(I)=m^*(I\cap E)+m^*(I\cap E^c)$. }
\item  {\color{red}对任意点集 $T\subseteq \mathbb{R}$ 都成立 $m^*(T)=m^*(T\cap E)+m^*(T\cap E^c)$. }
\end{enumerate} 

%\item  解答：


\end{itemize}

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\begin{frame}{3.2.3. 定义 }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是 $\mathbb{R}$ 的子集。什么时候称 $E$ 是勒贝格可测的？}

%\item  解答：
%如果对任意子集 $T\subseteq\mathbb{R}$ 都有 
%$$m^*(T)=m^*(T\cap E)+m^*(T\cap E^c), $$
%则称 $E$ 是勒贝格可测的。

\end{itemize}

\end{frame}

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\begin{frame}{3.2.4. 定理1 }

\begin{itemize}

\item  {\color{red}问题：集合 $E\subseteq \mathbb{R}$ 是勒贝格可测的，当且仅当对任意 $A\subseteq E$ 
与 $B\subseteq E^c$, 都有 $m^*(A\cup B) = m^*(A)+m^*(B)$.  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.2.5. 定理2 }

\begin{itemize}

\item  {\color{red}问题：集合 $E\subseteq \mathbb{R}$ 是勒贝格可测的，当且仅当 $E^c$ 是勒贝格可测的。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.2.6. 定理3 }

\begin{itemize}

\item  {\color{red}问题：设 $A,B$ 都是勒贝格可测的，则 $A\cup B$ 也是勒贝格可测的。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.2.7. 定理3+ }

\begin{itemize}

\item  {\color{red}问题：设 $A,B$ 都是勒贝格可测的，且 $A\cap B=\varnothing$. 则对于任意集合 $T$, 都有 
$$m^*(T\cap (A\cup B)) = m^*(T\cap A) + m^*(T\cap B). $$ 

 }

%\item  解答：


\end{itemize}

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\begin{frame}{3.2.8. 定理4 }

\begin{itemize}

\item  {\color{red}问题：设 $A,B$ 都是勒贝格可测的，则 $A\cap B$ 也是勒贝格可测的。 }

%\item  解答：


\end{itemize}

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\begin{frame}{3.2.9. 定理5 }

\begin{itemize}

\item  {\color{red}问题：设 $A,B$ 都是勒贝格可测的，则 $A- B$ 也是勒贝格可测的。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.2.10. 定理6 }

\begin{itemize}

\item  {\color{red}问题：设 $S_n$ 是一列互不相交的勒贝格可测集，则 $\cup_{n=1}^{\infty} S_n$ 也是勒贝格可测集，且有
$$m\left( \bigcup\limits_{n=1}^{\infty} S_n\right) = \sum_{n=1}^{\infty} m(S_n). $$ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.2.11. 定理7 }

\begin{itemize}

\item  {\color{red}问题：设 $S_n$ 是一列勒贝格可测集，则 $\cap_{n=1}^{\infty} S_n$ 也是勒贝格可测集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.2.12. 定理8 }

\begin{itemize}

\item  {\color{red}问题：设 $S_n$ 是一列递增的勒贝格可测集，记 $S=\cup_{n=1}^{\infty} S_n$. 
则 $$m(S)=\lim\limits_{n\to\infty} m(S_n). $$
  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.2.13. 定理9 }

\begin{itemize}

\item  {\color{red}问题：设 $S_n$ 是一列递降的勒贝格可测集，记 $S=\cap_{n=1}^{\infty} S_n$. 设 $m(S_1)<\infty$. 
则 $$m(S)=\lim\limits_{n\to\infty} m(S_n). $$ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.3.1. 可测集类 }

\begin{itemize}

\item  {\color{red}问题：证明下述结论： }
\begin{enumerate}
\item  若 $m^*(E)=0$, 则 $E$ 是勒贝格可测集。
\item  若 $m^*(E)=0$, 设 $F\subseteq E$, 则 $E$ 是勒贝格可测集。
\item  可数个零测度集的并集仍是零测度集。
\end{enumerate}

%\item  解答：


\end{itemize}

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\begin{frame}{3.3.2. 定理2 }

\begin{itemize}

\item  {\color{red}问题：证明（开、闭、半开半闭）区间都是勒贝格可测集，且测度为其长度。 }

%\item  解答：


\end{itemize}

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\begin{frame}{3.3.3. 定理3 }

\begin{itemize}

\item  {\color{red}问题：证明开集和闭集都是勒贝格可测集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.3.4.  }

\begin{itemize}

\item  {\color{red}问题：什么是 $\mathbb{R}^n$ 上的一个 $\sigma$-代数？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.3.5.  }

\begin{itemize}

\item  {\color{red}问题：设 $\Omega$ 是 $\mathbb{R}^n$ 上的一个 $\sigma$-代数。
什么是 $\Omega$ 上的一个（正）测度？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.3.6.  }

\begin{itemize}

\item  {\color{red}问题：什么是 $\mathbb{R}^n$ 上的博雷尔代数？  }

%\item  解答：


\end{itemize}

\end{frame}


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\begin{frame}{3.3.7. 定理4 }

\begin{itemize}

\item  {\color{red}问题：证明博雷尔集都是勒贝格可测集。 }

%\item  解答：


\end{itemize}

\end{frame}


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\begin{frame}{3.3.8. 定理5 }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是任意勒贝格可测集。则存在一列开集的交集 $G$ 使得 
$E \subseteq G$ 且 $m(G-E)=0$. 
}

%\item  解答：


\end{itemize}

\end{frame}


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\begin{frame}{3.3.9. 定理6 }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是任意勒贝格可测集。则存在一列闭集的并集 $F$ 使得 
$F\subseteq E$ 且 $m(E-F)=0$.  }

%\item  解答：


\end{itemize}

\end{frame}


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\begin{frame}{3.3.10. 定理7 }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是任意勒贝格可测集。则
\begin{eqnarray*}
m(E) &=& \inf \{ m(G): G \text{是开集, } E\subseteq G \} \\
&=& \sup \{ m(K): K \text{是紧集, } K\subseteq E \}.
\end{eqnarray*}

 }

%\item  解答：


\end{itemize}

\end{frame}


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\begin{frame}{3.4.1. 不可测集  }

\begin{itemize}

\item  {\color{red}问题：什么是勒贝格测度的平移不变性？证明勒贝格测度的平移不变性。 }

%\item  解答：


\end{itemize}

\end{frame}


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\begin{frame}{3.4.2.  }

\begin{itemize}

\item  {\color{red}问题：什么是勒贝格测度的反射不变性？ }

%\item  解答：


\end{itemize}

\end{frame}


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\begin{frame}{3.4.3.  }

\begin{itemize}

\item  {\color{red}问题：举例说明直线上存在勒贝格不可测集。 }

%\item  解答：


\end{itemize}

\end{frame}


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\begin{frame}{3.5.1. 习题1 }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq \mathbb{R}^n$ 是有界点集，证明 $m^*(E)<\infty$.  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.5.1. 习题2 }

\begin{itemize}

\item  {\color{red}问题：证明可数点集的外测度为零。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.5.1. 习题3 }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq\mathbb{R}$ 是有界点集。设 $m^*(E)>0$. 
设 $0<c<m^*(E)$, 证明存在子集 $E_1\subseteq E$ 使得 $m^*(E_1)=c$. 
 }

%\item  解答：


\end{itemize}

\end{frame}


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\begin{frame}{3.5.2. 习题4 }

\begin{itemize}

\item  {\color{red}问题：设 $S_1,S_2$ 是互不相交的可测集，设 $E_i\subseteq S_i$. 证明 
$$m^*(E_1\cup E_2) = m^*(E_1)+m^*(E_2). $$  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.5.2. 习题5 }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq\mathbb{R}^n$, 若 $m^*(E)=0$, 证明 $E$ 是勒贝格可测集。  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.5.2. 习题6 }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq\mathbb{R}^n$ 是可测集，若对任意区间 $I\subseteq\mathbb{R}^n$ 
有 $m(E\cap I)=0$, 证明 $m(E)=0$.  }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.5.3. 习题11 }

\begin{itemize}

\item  {\color{red}问题：设 $\{E_n\}$ 是一列可测集。证明 $\varliminf\limits_{n\to\infty} E_n$ 也是可测集，且有
$$m\left( \varliminf_{n\to\infty} E_n \right) \le \varliminf_{n\to\infty} m(E_n). $$
 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{3.5.4. 习题13 }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq [0,1]$ 是可测集，且 $m(E)=1$. 设 $A\subseteq [0,1]$ 也是可测集。证明
$m(E\cap A) = m(A).$  }

%\item  解答：


\end{itemize}

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\begin{frame}{3.5.5. 习题16 }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq \mathbb{R}$ 是可测集，设 $c\in\mathbb{R}$. 记 $cE=\{cx: x\in E\}$.证明 $cE$ 也是可测集。}

%\item  解答：


\end{itemize}

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